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Short Egyptian fractions

OpenAI groups related papers into a “family”. OpenAI’s catalogue describes this one:

Every rational \(a/b\) with \(1\le a\lt b\) is a sum of \(O(\log\log b)\) distinct positive unit fractions. The worst-case minimum number of terms has the same order, resolving Erdős’s conjecture on short Egyptian fractions. (OpenAI’s Lean note)

Lean is a language for mathematical statements and proofs that a computer can verify. The assessments below ask whether those statements express the result OpenAI describes here.

Both independent assessments find that the Lean statements express or imply the whole result described above. Read the assessments

It bears on Erdős problem #148, Erdős problem #293 and Erdős problem #304, as listed by Thomas Bloom, erdosproblems.com.

What the papers state

Corollary 1.3

In Short Egyptian fractions, 25 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every sufficiently large integer k,

\exp\bigl(\exp(k/600)\bigr)\le v(k)\le1+k^{2^{k-1}}.

More precisely,

\frac{\log2}{257} \le\liminf_{k\to\infty}\frac{\log\log v(k)}k \le\limsup_{k\to\infty}\frac{\log\log v(k)}k \le\log2.

Equivalently for the lower bound, every fixed c<\log2/257 satisfies v(k)\ge\exp(\exp(ck)) for every sufficiently large integer k. In particular \log\log v(k)=\Theta(k).

Theorem 1.1

In Short Egyptian fractions, 25 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There are absolute constants c_1,c_2>0 and b_0 such that, for every integer b\ge b_0,

c_1\log\log b\le N(b)\le c_2\log\log b.

In particular, the upper bound holds for every integer numerator 1\le a<b.

No Lean statement in the release is paired with this theorem.

Corollary 1.2

In Short Egyptian fractions, 25 September 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

There are absolute constants c,C>0 and k_0 such that, for every integer k\ge k_0,

ck\le\log\log F(k)\le Ck.

No Lean statement in the release is paired with this theorem.

The release does not say which paper theorem its catalogue describes. The statements above share the most wording with the description; the atlas has not reviewed that choice. Identifying it is listed under Questions awaiting review.

Reported proof verification
Two Lean statements in the release; independent groups report that Comparator accepted runs for all these files at the version shown here.
Papers and explanations
One current paper, readable here as a page beside its PDF. One explanation by others.

Papers

Each paper opens as a reading page in the browser, with the PDF beside it. The statements each paper opens with are on its reading page.

Short Egyptian fractions

25 September 2026, current edition.

Read it herePDF

We prove a conjecture of Erdős: for every sufficiently large integer b, every rational number \(a/b\) with \(1\le a\lt b\) is a sum of \(O(\log\log b)\) distinct positive unit fractions, with an absolute implied constant. This order is best possible when the numerator varies. We also show that both the number of expansions of 1 with exactly k distinct terms and the least integer at least 2 that never occurs as a denominator in such an expansion grow doubly exponentially in k: their double logarithms have order k.

Two Lean statements in this family

Choose one to see it beside the paper theorem it is paired with.

For a configured theorem, the Comparator program tests whether OpenAI’s separate proof proves the exact Lean statement using only the allowed axioms. Some statement files instead configure definitions for comparison by name and type. Acceptance does not establish that a formal statement means the same thing as the paper’s theorem.

EgyptianFractions

The paper states

Corollary 1.3

In Short Egyptian fractions, 25 September 2026

For every sufficiently large integer k,

\exp\bigl(\exp(k/600)\bigr)\le v(k)\le1+k^{2^{k-1}}.

More precisely,

\frac{\log2}{257} \le\liminf_{k\to\infty}\frac{\log\log v(k)}k \le\limsup_{k\to\infty}\frac{\log\log v(k)}k \le\log2.

Equivalently for the lower bound, every fixed c<\log2/257 satisfies v(k)\ge\exp(\exp(ck)) for every sufficiently large integer k. In particular \log\log v(k)=\Theta(k).

The words they share: k/600, exp, bound, log, integer, theta.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Short expansions, counting, and prescribed denominators

EgyptianFractions.lean, the release lists it as a main result.

theorem egyptian_length_is_minimum :    ∀ a b : ℕ, 1 ≤ a → a < b →  (∃ n : Fin (egyptianLength a b) → ℕ,    IsEgyptianExpansion ((a : ℚ) / (b : ℚ)) n) ∧  (∀ k : ℕ, (∃ n : Fin k → ℕ,    IsEgyptianExpansion ((a : ℚ) / (b : ℚ)) n) →    egyptianLength a b ≤ k) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

The file states eight more theorems: main_double_log_order, one_expansions_finite_and_bounded, counting_double_log_order, exact_marker_padding, every_exact_marker_occurs, quantitative_marked_length, missing_denominator_semantics, prescribed_denominator_corollary.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

ShortEgyptianFractions

The paper states

Proposition 2.1 (A dense set of short expansions)

In Short Egyptian fractions, 25 September 2026

There are absolute constants D_M>1, L>0, and S_0>1 with the following property. Put D_X=D_M+2 and D_C=4D_X. For every real S\ge S_0 and every integer C satisfying

e^{D_CS}\le C\le e^{2D_CS},

there are an integer M and a set G\subseteq\{1,\ldots,\lfloor X\rfloor\}, where X=e^{D_XS}, such that

e^S<M\le e^{D_MS}, \qquad \bigl|\{1,\ldots,\lfloor X\rfloor\}\setminus G\bigr|\le \frac X8.

For each u\in G, the fraction u/(MC) is a sum of at most L\log S unit fractions with integer denominators at least 2. Repetitions are allowed.

The words they share: real, expansions, log, integer, denominators, fraction.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Optimal order of the shortest Egyptian-fraction expansions

ShortEgyptianFractions.lean, the release lists it as a supporting statement.

theorem main :    (∀ a b : ℕ, 1 ≤ a → a < b → ∃ ns : List ℕ, IsExpansion a b ns) ∧    ∃ c₁ c₂ : ℝ, 0 < c₁ ∧ 0 < c₂ ∧ ∃ b₀ : ℕ, 2 ≤ b₀ ∧      ∀ b : ℕ, b₀ ≤ b →        c₁ * Real.log (Real.log (b : ℝ)) ≤ (maxMinLength b : ℝ) ∧        (maxMinLength b : ℝ) ≤ c₂ * Real.log (Real.log (b : ℝ)) := by  sorry

This file states the theorem with sorry, a placeholder, in place of its proof. OpenAI supplies a separate proof file. Comparator tests whether that proof proves the statement here.

two independent groups report that Comparator accepted runs for this Lean statement file: cert-machine; MathVet. Who ran the proof-verification program

  • cert-machine records it as “CERTIFIED”.
  • MathVet records it as “comparator-sandboxed-pass”.

Read the complete Lean statement

Other mathematical statements in OpenAI’s Lean library

Atlas found these propositions by scanning the text of the project’s Model.lean file. Each name links to its definition, with any associated statement file beside it. This list does not record running Lean to interpret or verify those definitions.

Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

MathVet and cert-machine agree on how much of the advertised result the Lean statements express.

MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: None; unit-fraction expansions use plain List ℕ/Fin k → ℕ with strict monotonicity and rational sums.

“full”, in their words: The challenge statements imply the headline claim after unpacking definitions or one routine step.

Their method: An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.

Read the original assessment Read MathVet’s method

cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

Read the original assessment Read cert-machine’s method

OpenAI’s own note on what its Lean covers

An Egyptian-fraction expansion writes a rational number as a sum of distinct unit fractions. The formalization proves that every $a/b$ with $1\le a<b$ has such an expansion and that the largest minimum length at denominator $b$ is $\Theta(\log\log b)$. If $F(k)$ counts exact $k$-term expansions of $1$, it also proves $\log\log F(k)=\Theta(k)$ and bounds every denominator in such an expansion.

Every integer $m\ge2$ occurs as a denominator in an expansion of $1$, with length eventually at most $(257/\log2+\varepsilon)\log\log m$ for every $\varepsilon>0$; padding can preserve that denominator. If $v(k)$ is the least integer at least $2$ absent from all exact $k$-term expansions of $1$, then eventually $e^{e^{k/600}}\le v(k)\le1+k^{2^{k-1}}$, and $\liminf_{k\to\infty}\log\log v(k)/k\ge\log2/257$.

AI-drafted comparisons, with no recorded human review

  • Short expansions, counting, and prescribed denominators EgyptianFractions

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem (the largest minimum expansion length at denominator b is of order log log b) together with the counting corollary and the results on prescribed denominators.

  • Optimal order of the shortest Egyptian-fraction expansions ShortEgyptianFractions

    Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the largest minimum length of an Egyptian-fraction expansion at denominator b is of order log log b.

Who ran the proof-verification program?

These groups report running Comparator with the release’s Lean statement files. For a configured theorem, it verifies OpenAI’s separate proof against the exact statement; some files instead configure definitions for comparison by name and type. Each row is the named group’s report. Atlas has not repeated the runs.

These reports do not list the software versions, proof-verification programs, axioms, time or memory in separate fields. Any such details in a report’s prose are reproduced below.

Short expansions, counting, and prescribed denominators EgyptianFractions

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Optimal order of the shortest Egyptian-fraction expansions ShortEgyptianFractions

Two reports, all at the release’s recorded version.

Reported byTheir wordTheir accountDateOriginal report
cert-machine“CERTIFIED”Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.10 October 2026Original report
MathVet“comparator-sandboxed-pass”Re-run through Comparator in a sandboxed virtual machine: accepted.8 October 2026Original report

Explanations by others

Explanations of this family written elsewhere, by the people named. Each is their own reading of the mathematics.

  • Woett, Short Egyptian fractions (a simplified write-up), 10 October 2026. Source

    A simplified human-written proof gives an explicit constant of about 23 for the number of unit fractions against about 371 in the manuscript, and the same argument yields the two related Erdős problems.

What these papers cite in the release, and what cites them

These links come from the papers’ bibliographies. A citation records that one paper cites another; how each cited result is used has not been examined here.

Papers elsewhere in the release that this family’s papers cite

This family’s papers cite no paper elsewhere in the release.

Papers elsewhere in the release that cite this family’s papers

No paper elsewhere in the release cites this family’s papers.

Follow-up work by others

  • The Justin Sun Prize, The Justin Sun Prize: problem bank and award records, 11 October 2026. Source

    JSP-000255 lists the release’s claim as Solved with independent mathematical review pending; Lean proof column: reported complete, independent verification pending.

  • Woett, Short Egyptian fractions (a simplified write-up), 9 October 2026. Source

    Simplified proof with constant 16/log 2 for #304 and consequences for #148 and #293, announced in Bloom’s comment thread.

References outside the release

Each paper’s outside references are listed with it on its reading page, where each citation is placed. Every outside work cited in the release: References outside the release

Explain a result, compare statements or propose a correction

Atlas has no contributed explanation for this family yet. You can offer one with a GitHub account. You receive named credit and disclose any AI assistance.

Each button opens a GitHub form with this page and its source version filled in. Opening the form does not reserve the work.

Offer to write an explanationCompare one Lean statement with its paper: EgyptianFractions, ShortEgyptianFractionsSuggest a source attributionSuggest a change

What people have said

Commentary on this family by others, dated and credited. It is commentary, not a finding.

  • An unsigned public tier list, the OpenAI Math Tier List, classes it as a proof and rates its significance, if it holds up, as “Solid publication, could lead to tenure”. The page says that nothing on it is verified.

    Author not stated. Source

  • Moritz Firsching, anxkhn, Paul Lezeau, Formal Conjectures: mark problems solved in the OpenAI math release and link formal proofs, 11 October 2026. Source

    A Formal Conjectures maintainer, after source inspection, links the short Egyptian fractions result to Erdos problem 304.

  • Woett, Short Egyptian fractions (a simplified write-up), 10 October 2026. Source

    The same author reports that the manuscript proves the main bound in a hard twenty-page argument and then uses a different, easier argument for one corollary, and that the formalisations of two of the problems go through the hard proof while the third modifies the easy one to avoid a deep prime number theorem.

  • eigendash, eigendash/openai-math-visualised: numerical probes of seven result families, 8 October 2026. Source

    For denominators b up to 220, the shortest Egyptian-fraction length divided by log log b stays between 2.43 and 4.80, consistent with the predicted order log log b over that range.

  • Thomas Bloom, Erdős problems in the OpenAI proofs, 8 October 2026. Source

    Bloom lists Erdős 148, 293 and 304 as three unit fraction results from one 33-page PDF, names the previous best bounds, and links one formalisation.

Questions awaiting review

Each item identifies a statement, definition or report that needs further review. Its link opens a GitHub form with the source version already recorded.

Read the Lean statement against the paper

  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form
  • An AI agent drafted an assessment of which part of the paper this Lean statement expresses. No named human review is recorded. Open the form

Identify the theorem the catalogue describes

  • Family 025 No single paper statement clearly matches OpenAI’s description by wording, and no review identifying the intended theorem is recorded. Open the form

Compare the proposed paper and Lean matches

  • Family 025 Atlas records no review of whether each linked Lean statement and paper theorem mean the same thing. Open the form